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Misc 12 Solve tan-1 (1 − x)/(1 + x) = 1/2 tan-1 x, (x > 0) tan-1 (1 − x)/(1 + x) = 1/2 tan-1 x 2 tan-1 ((1 − x)/(1 + x)) = tan-1 x tan-1 [(2 ((1 − 𝑥)/(1 + 𝑥)))/(1 − ((1 − 𝑥)/(1 + 𝑥 ))^2 )] = tan-1 x We know that 2 tan-1 x = tan-1 ((𝟐𝒙 )/(𝟏 − 𝐱^𝟐 )) Replacing x by (1 − 𝑥)/(1 + 𝑥) tan-1 [((2 (1 − 𝑥))/((1 + 𝑥)))/(((1 + 𝑥)2 − ( 1 −𝑥 )2)/(1 + 𝑥 )^2 )] = tan-1 x tan-1 [(2 (1 − 𝑥))/((1 + 𝑥)) × ((1 + 𝑥))/((1 + 𝑥)2 − (1 − 𝑥)2)] = tan-1 x tan-1 [(2 (1 − 𝑥) (1 + 𝑥))/((1 + 𝑥)2 − (1 − 𝑥)2)] = tan-1 x Using (a + b) (a – b) = a2 – b2 tan-1 [ (2 (1 − 𝑥2) )/((1 + 𝑥 + 1 − 𝑥) (1+ 𝑥 − 1 + 𝑥) )] = tan-1 x tan-1 [ (2 (1 − 𝑥2) )/((1 +1 − 𝑥 − 𝑥) (𝑥 + 𝑥 − 1 + 1) )] = tan-1 x tan-1 [(2 (1 − 𝑥2))/(4 (1) (𝑥) )] = tan-1 x tan-1 [(1 − 𝑥2)/2𝑥] = tan-1 x Comparing values (1 − 𝑥2)/2𝑥 = x 1 – x2 = 2x2 1 – x2 – 2x2 = 0 1 – 3x2 = 0 3x2 = 1 x2 = 1/3 x = ± 1/√3 x = (− 1 )/√3 is not possible because it is Given that x > 0 Hence, x = ( 𝟏 )/√𝟑

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.